Hawkes process

In probability theory and statistics, a Hawkes process, named after Alan G. Hawkes, is a kind of self-exciting point process.^[1] It has arrivals at times ${\textstyle 0<t_{1}<t_{2}<t_{3}<\cdots }$ where the infinitesimal probability of an arrival during the time interval ${\textstyle [t,t+dt)}$ is

\lambda _{t}\,dt=\left(\mu (t)+\sum _{t_{i}\,:\,t_{i}\,<\,t}\phi (t-t_{i})\right)\,dt.

The function ${\textstyle \mu }$ is the intensity of an underlying Poisson process. The first arrival occurs at time ${\textstyle t_{1}}$ and immediately after that, the intensity becomes ${\textstyle \mu (t)+\phi (t-t_{1})}$ , and at the time ${\textstyle t_{2}}$ of the second arrival the intensity jumps to ${\textstyle \mu (t)+\phi (t-t_{1})+\phi (t-t_{2})}$ and so on.^[2]

During the time interval ${\textstyle (t_{k},t_{k+1})}$ , the process is the sum of ${\textstyle k+1}$ independent processes with intensities ${\textstyle \mu (t),\phi (t-t_{1}),\ldots ,\phi (t-t_{k}).}$ The arrivals in the process whose intensity is ${\textstyle \phi (t-t_{k})}$ are the "daughters" of the arrival at time ${\textstyle t_{k}.}$ The integral $\int _{0}^{\infty }\phi (t)\,dt$ is the average number of daughters of each arrival and is called the branching ratio. Thus viewing some arrivals as descendants of earlier arrivals, we have a Galton–Watson branching process. The number of such descendants is finite with probability 1 if branching ratio is 1 or less. If the branching ratio is more than 1, then each arrival has positive probability of having infinitely many descendants.

Applications

Hawkes processes are used for statistical modeling of events in mathematical finance,^[3] epidemiology,^[4] earthquake seismology,^[5] and other fields in which a random event exhibits self-exciting behavior.^[6]^[7]

References

^ Laub, Patrick J.; Lee, Young; Taimre, Thomas (2021). The Elements of Hawkes Processes. doi:10.1007/978-3-030-84639-8. ISBN 978-3-030-84638-1. S2CID 245682002.
^ Hawkes, Alan G. (1971). "Spectra of some self-exciting and mutually exciting point processes". Biometrika. 58 (1): 83–90. doi:10.1093/biomet/58.1.83. ISSN 0006-3444.
^ Hawkes, Alan G. (2018). "Hawkes processes and their applications to finance: a review". Quantitative Finance. 18 (2): 193–198. doi:10.1080/14697688.2017.1403131. ISSN 1469-7688. S2CID 158619662.
^ Rizoiu, Marian-Andrei; Mishra, Swapnil; Kong, Quyu; Carman, Mark; Xie, Lexing (2018). "SIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Populations". Proceedings of the 2018 World Wide Web Conference on World Wide Web - WWW '18. pp. 419–428. arXiv:1711.01679. doi:10.1145/3178876.3186108. S2CID 195346881.
^ Kwon, Junhyeon; Zheng, Yingcai; Jun, Mikyoung (2023). "Flexible spatio-temporal Hawkes process models for earthquake occurrences". Spatial Statistics. 54: 100728. arXiv:2210.08053. Bibcode:2023SpaSt..5400728K. doi:10.1016/j.spasta.2023.100728. S2CID 252917746.
^ Tench, Stephen; Fry, Hannah; Gill, Paul (2016). "Spatio-temporal patterns of IED usage by the Provisional Irish Republican Army". European Journal of Applied Mathematics. 27 (3): 377–402. doi:10.1017/S0956792515000686. ISSN 0956-7925. S2CID 53692006.
^ Laub, Patrick J.; Taimre, Thomas; Pollett, Philip K. (2015). "Hawkes Processes". arXiv:1507.02822 [math.PR].

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Hawkes process

Applications

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References

Further reading